The propagation of a cylindrical shock wave in rotating medium with azimuthal magnetic field under the action of monochromatic radiation using a method of group invariance is investigated. To derive similarity solutions as well as exact solutions, the group invariance technique is used. All classes of the solutions depending on the absorption coefficient are discussed by considering absorption coefficient to be variable or constant. A similarity solution is obtained, when the absorption coefficient is assumed to be variable. Two cases of solutions with a power law shock path are obtained by the different choices of arbitrary constants involving in the infinitesimal generators of the Lie group of transformations. To obtain the similarity solution in the case of the power law shock path, the density, magnetic field, axial and azimuthal velocity components are assumed to be varying and obeying power laws in the undisturbed medium. It is observed that with increase in the values of Alfven Mach number, adiabatic exponent and rotational parameter, shock strength decreases. The effects of variation of magnetic field strength, adiabatic exponent, rotational parameter and initial magnetic field variation index on the flow variables and on shock waves are analyzed graphically. Also, all classes of exact solutions are obtained by considering a constant absorption coefficient.

The main purpose of this investigation is to discuss the entire class of similarity solutions for adiabatic flow behind the cylindrical shock wave in a dusty gas (a mixture of non-ideal gas and small solid particles) using the Lie group invariance method. The group invariance method is used to derive the similarity transformations and the similarity variable, which transforms the set of partial differential equations into a set of ordinary differential equations. The obtained set of ordinary differential equations is solved numerically using the Runge–Kutta method of fourth order in Mathematica software. It is found that the shock wave decays with the variation of the gas non-idealness parameter. Also, the shock strength enhanced with an increase in the ratio of the specific heat of the solid particles to the specific heat of the gas at constant volume or the initial azimuthal velocity variation exponent or the ratio of the density of solid particles to the initial density of the gas. The behavior of the flow variables behind the shock wave is analyzed graphically. This study may be helpful in understanding the astrophysical phenomena such as supernovae explosion, in a dusty gas environment.

In this article, the similarity solution using the Lie group theoretic method for unsteady isothermal and adiabatic flows behind cylindrical or spherical shock wave in a mixture of self-gravitating real gas and small solid particles is discussed. The Lie group theoretic method gives out different cases, i.e., exponential law and power-law shock paths. The similarity solution exists only when the shock path varies according to an exponential law. The dispersal of the flow variables with the variations of the solid particles’ mass concentration in the mixture kp, gravitational parameter g0, the ratio of the density of solid particles to the initial density of the gas μ1, non-idealness parameter b¯ and the geometry index ν are discussed graphically. It is found that an increase in the value of μ1 or g0 or ν leads to an increase in the shock strength, but the shock wave decay with an increase in b¯. A comparison between the solutions for cylindrical and spherical symmetry, and for adiabatic and isothermal flows is also made.

We have studied the one dimensional unsteady isothermal and adiabatic flows behind the shock waves in a non-ideal dusty gas under the effect of magnetic field using Lie group invariance method. This method is used to get the infinitesimal generators of the Lie algebra. The concept of optimal systems is used for minimizing the group-invariant solutions. The optimal system of subalgebra is used to obtain the similarity variables and similarity transformation which transform the set of partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). Consequently, the system of ODEs in different cases is solved by using Runge–Kutta (R-K) method of forth order to derive the similarity solution in the cases of isothermal and adiabatic flows. The detailed discussions are illustrated by graph through numerical calculation in the case of power-law shock path. It is shown that the shock wave strength has decaying effects with an increase in the values of Alfven Mach number, adiabatic exponent of the gas, and gas non-idealness parameter. Also, the shock strength increases with an increment in the values of mass concentration of solid particles, magnetic field variation exponent, the ratio of density of solid particles to the gas initial density, and by changing the geometry from cylindrical to spherical.

This paper concerns with the similarity solutions using Lie group theoretic method for one-dimensional unsteady isothermal flow behind the shock wave in the mixture of non-ideal gas and small solid particles in rotating medium. Group theoretic technique brings the different possible cases of potential solutions with the power law, particular case of power law and exponential law shock paths with the different choices of constants present in the generators of the Lie group of transformations. Similarity solutions for non-ideal dusty gas (a mixture of non-ideal gas and small solid dust particles) exist if the density of the ambient medium is constant. Solutions are obtained for both exponential law and power law shock paths. The effects of the variation of the physical parameters on the flow variables distribution behind the shock wave front, and on the shock are discussed. The shock strength increases with an increase in the ratio of the specific heat of the solid particles to the specific heat of the gas at constant volume [Formula: see text], the initial azimuthal velocity variation index [Formula: see text], and the ratio of the density of solid particles to the initial density of the gas [Formula: see text]. The shock wave decays with an increase in the value of gas non-idealness parameter [Formula: see text].

The similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the case of isothermal and adiabatic flows are obtained. All possible cases of similarity solutions are discussed using the Lie group analysis for the isothermal and adiabatic flows. The arbitrary constants involved in the generators of local Lie groups bring various possible cases of solutions with exponential law and power law shock paths. Similarity solution for isothermal and adiabatic flows with power law shock path is discussed in detail. The density of ambient medium is taken to be constant. The axial and azimuthal fluid velocities and magnetic field in the ambient medium are assumed to be varying according to power law. The effect on shock wave strength and that on the flow variables due to variation of the Alfven Mach number, adiabatic index of the gas, non-idealness parameter, rotational parameter and initial magnetic field variation exponent are investigated. It is found that these parameters have decaying effects on shock wave. The obtained results in the case of isothermal and adiabatic flows are also compared with each other.

In this paper, we seek the exact and numerical solutions using Lie group analysis for one dimensional unsteady adiabatic flow in a self-gravitating ideal gas behind a cylindrical shock wave with axial magnetic field. With the help of Lie group analysis, the one-dimensional optimal system of sub-algebra is obtained for the system of equations of motion. With the help of optimal classes of infinitesimal generators, we constructed the similarity variable and transformation of the flow variables, which convert the system of partial differential equations into system of ordinary differential equations. In three particular cases, we have derived a general framework to solve the fundamental equations and exact feasible solutions are obtained. In two cases, the similarity solutions with exponential law and power law shock path are discussed. The similarity solution is obtained using numerical method in the case of power law shock path. It is obtained that the increase in the values of magnetic field strength and adiabatic index have the decaying effect on the shock wave. Also, increase in the strength of shock wave is witnessed with the rise in the gravitation parameter value. The effects of variation of magnetic field strength, adiabatic exponent and gravitational parameter on the flow variables are analyzed graphically.

In this paper, we have studied the propagation of cylindrical shock waves in a self-gravitating perfect gas under the influence of azimuthal magnetic field. The method of Lie group invariance is used to construct some special class of self-similar solutions in the presence of the azimuthal magnetic field. The different cases of solutions with a power law and exponential law shock paths are obtained with the choice of arbitrary constants appearing in the expressions for the infinitesimal generators. The similarity solution for cylindrical shock wave with power law shock path is discussed in detail. The effects of variation of Alfven-Mach number, gravitation parameter, initial density variation index and adiabatic exponent on the flow variables are analyzed graphically. It is obtained that the increase in the values of Alfven-Mach number, gravitation parameter and adiabatic exponent have decaying effect on the shock strength. Also, the shock strength increases with an increase in the values of initial density variation index. A comparison is also made between the solutions in gravitating and non-gravitating cases in the presence of magnetic field.